Algebraic Geometry

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Description

Objective: to offer you a broad perspective on algebraic geometry and some of the tools it provides. Our focus will be on the theory of schemes, but I'll be focusing on the concepts and examples that will be of interest to a broad range of students.

Textbook: [V] Ravi Vakil, The Rising Sea. I consider this the new standard algebraic geometry textbook. Reading this is worth your time. The lectures will be more fun if you can look over some of the material beforehand.

Location: Lectures will be in Bayes 5.46, Monday 1-3 pm. Tutorials to be determined in coordination with the tutor (Karim Rega)

Lecture plan: If you're reading this in Week n, then I estimate the accuracy of the topics listed for Week m to be around max(1, (4/5)^{m-n}).

Week 1. 7 Oct. Spec. The Zariski topology. Generic points. Some words about sheaves. [V, Ch. 2 & 3]
Week 2. 14 Oct. More words about sheaves. Locally ringed spaces. Schemes. Morphisms. Nilpotents. [V, Ch. 4 & 7]
Week 3. 21 Oct. Pullbacks. Open & closed. Functor of points. [V, Ch. 8-10]
Week 4. 28 Oct. Projective space. Quasicoherent modules. [V, Ch. 6 & 14]
Week 5. 4 Nov. Some properties of schemes and morphisms. Topological properties. Reducedness. [V, Ch. 5]
Week 6. 11 Nov. Separatedness. Properness. [V, Ch. 11]
Week 7. 18 Nov. Vector bundles. Line bundles. [V, Ch. 15 &16]
Week 8. 25 Nov. Cohomology [V, Ch. 18]
Week 9. 2 Dec. Kähler differentials [V, Ch. 21]
Week 10. 9 Dec. Dimension & smoothness. [V, Ch. 12 & 13]